On partial orderings having precalibre-$\aleph _1$ and fragments of Martin's axiom

Joan Bagaria ¹   ; Saharon Shelah ²

¹ ICREA (Institució Catalana de Recerca i Estudis Avançats) and Departament de Lògica, Història i Filosofia de la Ciència Universitat de Barcelona Montalegre 6 08001 Barcelona, Catalonia, Spain
² Einstein Institute of Mathematics The Hebrew University of Jerusalem Edmond J. Safra Campus Givat Ram, Jerusalem 91904, Israel

Fundamenta Mathematicae, Tome 232 (2016) no. 2, pp. 181-197

Résumé

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-$\aleph _1$, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for $\sigma $-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for $\sigma $-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprāns and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-$\aleph _1$ property of a partial ordering while preserving its ccc-ness.

Détail

DOI : 10.4064/fm232-2-6

Keywords: define countable antichain condition ccc property partial orderings weaker precalibre aleph martins axiom restricted class partial orderings have property does imply martins axiom sigma linked partial orderings yields solution old question first author about relative strength martins axiom sigma centered partial orderings together assertion every aronszajn tree special answer question stepr watson showing forcing preserves cardinals destroy precalibre aleph property partial ordering while preserving its ccc ness